I used to do it comfortably enough in a 3.8m wood floor avon with a 15hp on it, out of lymington and over to the folly inn etc. A 3.4 with a ten would probably be fine too as a minimum. If you mean to cross past hurst point or go round the needles I wouldn't skimp on power or size, and choose your weather/ tide carefully.

Wondering what the minimum size of rib/sib and engine size would be recommended for a trip from Lymington to the Isle of Wight?

Where do you want to go? A short blast over to Yarmouth can be tackled in appropriate conditions/visibility in a fairly small craft - with normal caveats about safety and experience etc... we go over in a 4M Zodiac with a 40hp motor quite often. However the sea conditions out of Lymington can get very choppy, with the winds funneling through the Hurst channel, and especially when you have wind over tide, so pick your weather carefully. If you plan to go along the IoW coast towards Cowes, then I tend to cross the Solent quite early, and then hug the shore.. but in a small boat, check the tides, and go with them. Another option for Cowes is hug the New Forest shoreline and then cross the Solent at the narrow point at Lepe. I'm sure lots of folk here will have there own suggestions too!

As said above it all depends on weather and tides. When I was a kid we did it in a 3.1m Zosiac, 4hp outboard and 4 people, it was flat calm and we used the tides to help us both ways. Yesterday I had to turn back in a 8m sportscruiser ! I do agree with above, 3.4m and 10hp as a minimum. Choose your conditions very carefully, carry safety equipment ( flares, vhf, lifejackets, anchor) make sure someone knows you are route and go for it . Make sure you watch out for the ferries, they travel much faster than you think.

I regularly cross to the Island in my Avon Rover 340, previously fitted with a Tohatsu 18hp and, more recently, with the Suzuki 25hp. Agree that you need to pick the right weather (paying particular attention to the forecast for your likely return time). My rule of thumb is 'if in doubt, don't do it'.